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We will denote by x and y the populations of the prey and predator, respectively, at time t. In constructing a model of the interaction of the two species, we make the following assumptions:
1. In the absence of the predator the prey grows at a rate proportional to the current population; thus dx/dt = ax, a > 0, when y = 0.
2. In the absence of the prey the predator dies out; thus dy/dt =cy, c > 0, when
x = 0.
3. The number of encounters between predator and prey is proportional to the product of their populations. Each such encounter tends to promote the growth of the predator and to inhibit the growth of the prey. Thus the growth rate of the predator is increased by a term of the form y xy, while the growth rate of the prey is decreased by a term axy, where y and a are positive constants.
As a consequence of these assumptions, we are led to the equations
dx/ dt = ax axy = x (a a y), dy/dt = -cy + y xy = y(c + y x).
The constants a, c, a, and y are all positive; a and c are the growth rate of the prey
and the death rate of the predator, respectively, and a and y are measures of the effect
2AlanL. Hodgkin (1914-1998) and Andrew F. Huxley (1917) were awarded the Nobel prize in physiology and medicine in 1963 for their work on the excitation and transmission of neural impulses, first published in 1952, when they were at Cambridge University.
Chapter 9. Nonlinear Differential Equations and Stability
of the interaction between the two species. Equations (1) are known as the Lotka-Volterra equations. They were developed in papers by Lotka3 in 1925 and by Volterra4 in 1926. Although these are rather simple equations, they do characterize a wide class of problems. Ways of making them more realistic are discussed at the end of this section and in the problems. Our goal here is to determine the qualitative behavior of the solutions (trajectories) of the system (1) for arbitrary positive initial values of x and y. We do this first for a specific example and then return to the general equations (1) at the end of the section.
Discuss the solutions of the system
dx/dt = x(1 - 0.5y) = x - 0.5xy,
dy/dt = y(-0.75 + 0.25x) = 0.75y + 0.25xy
for x and y positive.
The critical points of this system are the solutions of the algebraic equations
x(1 0.5 y) = 0, y(0.75 + 0.25x) = 0,
namely, the points (0, 0) and (3, 2). Figure 9.5.1 shows the critical points and a direction field for the system (2). From this figure we conclude tentatively that the trajectories in the first quadrant may be closed curves surrounding the critical point (3, 2).
Next we examine the local behavior of solutions near each critical point. Near the origin we can neglect the nonlinear terms in Eqs. (2) to obtain the corresponding linear system
The eigenvalues and eigenvectors of Eq. (4) are
D = 1,
r2 = -0.75,
so its general solution is
e + c2
Thus the origin is a saddle point of both the linear system (4) and of the nonlinear system (2), and therefore is unstable. One pair of trajectories enters the origin along the y-axis; all other trajectories depart from the neighborhood of the origin.
3Alfred J. Lotka (1880 -1949), an American biophysicist, was born in what is now the Ukraine, and was educated mainly in Europe. He is remembered chiefly for his formulation of the Lotka-Volterra equations. He was also the author, in 1924, of the first book on mathematical biology; it is now available as Elements of Mathematical Biology (New York: Dover, 1956).
4Vito Volterra (1860-1940), a distinguished Italian mathematician, held professorships at Pisa, Turin, and Rome. He is particularly famous for his work in integral equations and functional analysis. Indeed, one of the major classes of integral equations is named for him; see Problem 20 of Section 6.6. His theory of interacting species was motivated by data collected by a friend, D Ancona, concerning fish catches in the Adriatic Sea. A translation of his 1926 paper can be found in an appendix to R. N. Chapman, Animal Ecology with Special Reference to Insects (New York: McGraw-Hill, 1931).
9.5 Predator-Prey Equations 505
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